What x values is the function concave down iff(x) 15x 5x ...

What x values is the function concave down ifÂ Â f(x)=15xâ…”+5x ?

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Advance Math

What x values is the function concave down ifÂ Â f(x)=15xâ…”+5x ?

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Note thatÂ f(0)=0

and checking for critical points by taking the derivative and setting toÂ 0

we get

f'(x)=10x-â…“+5=0

or 10xâ…“=âˆ’5

which simplifies (ifÂ x<>0) to

xâ…“=âˆ’2

â†’Â x=âˆ’8

AtÂ x=âˆ’8

f(âˆ’8)=15(âˆ’8)23+5(âˆ’8)

=15(âˆ’2)2+(âˆ’40)

=20

Since (âˆ’8,20) is the only critical point (other than (0,0) )

andÂ f(x)Â decreases fromÂ x=âˆ’8Â toÂ x=0

it follows thatÂ f(x)Â decreases on each side of (âˆ’8,20), so

f(x)Â is concave downward whenÂ x<0.

WhenÂ x>0Â we simply note that

g(x)=5xÂ is a straight line and

f(x)=15xâ…”+5x remains a positive amount (namely 15xâ…”above that line

thereforeÂ f(x)Â is not concave downward forÂ x>0.

Â .

f(x)is concave downward whenÂ Â x<0Â .

WhenÂ Â x>0Â we simply note that

g(x)=5xÂ is a straight line and

f(x)=15xâ…”+5xÂ

Â remains a positive amount (namelyÂ Â 15xâ…”Â above that line thereforeÂ Â

f(x)is not concave downward forÂ Â x>0

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What x values is the function concave down ifÂ Â f(x)=15xâ…”+5x ?

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What x values is the function concave down ifÂ Â f(x)=15xâ…”+5x ?

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